Growth Factor Distribution ModelTrip Distribution

Theory

Trip Distribution

The generated trips from each zone is distributed to all other zones based on the choice of destination, this is called trip distribution which forms the second stage of travel demand modeling. This step matches trip maker' origins and destinations to develop a "trip table" matrix that displays the number of trips going from each origin to each destination. A typical trip table is represented below.

Origin/Destination	1	2	3	j	$\sum \limits_{j} T_{ij}$
1	$T_{11}$	$T_{12}$	$T_{13}$	$T_{1j}$	$O_{1}$
2	$T_{21}$	$T_{22}$	$T_{23}$	$T_{2j}$	$O_{2}$
3	$T_{31}$	$T_{32}$	$T_{33}$	$T_{3j}$	$O_{3}$
4	$T_{i1}$	$T_{i2}$	$T_{i3}$	$T_{ij}$	$O_{i}$
$\sum \limits_{i} T_{ij}$	$D_{1}$	$D_{2}$	$D_{3}$	$D_{j}$	$T = \sum \limits_{ij} T_{ij}$

Where $T_{ij}$ is the number of trips between origin i and destination j, $O_{i}$ is total number of trips originating from zone i and $D_{j}$ is the total number of trips attracted to zone j, T is total trips. Note that the practical value of trips on the diagonal, e.g. from zone 1 to zone 1, is zero since no intra-zonal trip occurs.

There are a number of methods to distribute trips among destinations; and two such methods are growth factor model and gravity model.

Growth Factor Model

Growth factor model is one of the methods among the number of methods to distribute trips among destinations. Growth factor model is a method which responds only to relative growth rates at origins and destinations and this is suitable for short term trend extrapolation.

Types of growth factor models

• Uniform Growth factor method
• Singly constrained growth factor method
• Doubly Constraint growth factor method

Uniform Growth Factor Method

If the only information available is about a general growth rate for the whole of the study area, then we can only assume that it will apply to each cell in the matrix, that is a uniform growth rate. The equation can be written as: Tij = f x tij

Where f is the uniform growth factor tij is the previous total number of trips, Tij is the expanded total number of trips. Advantages are that they are simple to understand, and they are useful for short-term planning.

Example

Trips originating from zone 1, 2, 3 of a study area are 78, 92 and 82 respectively and those terminating at zones 1,2,3 are given as 88,96 and 78 respectively. If the growth factor is 1.3 and the cost matrix is as shown below, find the expanded origin-constrained growth trip table.

Zone	1	2	3	$O_{i}$
1	20	30	28	78
2	36	32	24	92
3	22	34	26	82
$D_{j}$	88	96	78	252

Solution

Given growth factor = 1.3, Therefore, multiplying the growth factor with each of the cells in the matrix gives the solution as shown below.

Zone	1	2	3	$O_{i}$
1	26	39	36.4	101.4
2	46.8	41.6	31.2	101.4
3	28.6	44.2	33.8	106.2
$D_{j}$	101.4	124.8	124.8	327.6

Singly Constrained Growth Factor method

This method is used when the expected growth of either trips originated or trips destined are available. An example below illustrates how to solve for such models

Example

Consider the following matrix.

Zones	1	2	3	4	Total	Target
1	5	50	100	200	355	400
2	50	5	100	300	455	460
3	50	100	5	1000	255	400
4	100	200	250	20	570	702
Total	205	355	455	620	1635	1962

Method

Zones	1	2	3	4	Total	Target
1	5.6	56.3	112.7	225.4	400	400
2	50.5	5.1	101.1	303.3	460	460
3	78.4	156.9	7.8	156.9	400	400
4	123.2	246.3	307.9	24.6	702	702
Total	257.7	464.6	529.5	710.2	1962	1962

Doubly Constraint Growth Factor method Example

This method is used when the growth of trips originated and distributed for each zone is available. Thus two growth factor sets are available for each zones. Consequently there are two constraints and such a model is called as Double Constraint Growth Factor model

An example below illustrates how to solve for such models

O-D Matrix For Base Year

Zone	1	2	3	4	5	Total	Target
1	10	15	20	5	0	50	150
2	5	2	32	12	32	83	120
3	2	3	3	14	20	42	75
4	1	5	1	4	5	16	45
5	0	4	3	5	5	17	120
Total	18	29	59	40	62	208	---
Target	48	75	48	150	189	---	510

Doubly Constraint Growth Factor Matrix For Future Year

Zone	1	2	3	4	5	Current Origins Total	Origins Total Future year
1	30	45	60	15	0	150	150
2	7.229	2.892	46.265	17.349	46.265	120	120
3	3.571	5.357	5.357	25	35.714	75	75
4	2.813	14.063	2.813	11.25	14.063	45	45
5	0	28.235	21.176	35.294	35.294	120	120
Current Destinations Total	43.613	95.547	135.611	103.894	131.336
Destinations Total Future year	48	75	48	150	189

Doubly Constraint Growth Factor Matrix For Future Year

Zone	1	2	3	4	5	Current Origins Total	Origins Total Future year
1	30	45	60	15	0	150	150
2	7.229	2.892	46.265	17.349	46.265	120	120
3	3.571	5.357	5.357	25	35.714	75	75
4	2.813	14.063	2.813	11.25	14.063	45	45
5	0	28.235	21.176	35.294	35.294	120	120
Current Destinations Total	43.613	95.547	135.611	103.894	131.336
Destinations Total Future year	48	75	48	150	189

Doubly Constraint Growth Factor Matrix For Future Year

Zone	1	2	3	4	5	Current Origins Total	Origins Total Future year
1	30	45	60	15	0	150	150
2	7.229	2.892	46.265	17.349	46.265	120	120
3	3.571	5.357	5.357	25	35.714	75	75
4	2.813	14.063	2.813	11.25	14.063	45	45
5	0	28.235	21.176	35.294	35.294	120	120
Current Destinations Total	43.613	95.547	135.611	103.894	131.336
Destinations Total Future year	48	75	48	150	189

Doubly Constraint Growth Factor Matrix For Future Year

Zone	1	2	3	4	5	Current Origins Total	Origins Total Future year
1	30	45	60	15	0	150	150
2	7.229	2.892	46.265	17.349	46.265	120	120
3	3.571	5.357	5.357	25	35.714	75	75
4	2.813	14.063	2.813	11.25	14.063	45	45
5	0	28.235	21.176	35.294	35.294	120	120
Current Destinations Total	43.613	95.547	135.611	103.894	131.336
Destinations Total Future year	48	75	48	150	189

Similarly continuing till 4th iteration we will get the final result with accuracy level 3% of each individual cell as shown below.

Final Result

Doubly Constraint Growth Factor Matrix For Future Year

Zone	1	2	3	4	5	Current Origins Total	Origins Total Future year
1	30	45	60	15	0	150	150
2	7.229	2.892	46.265	17.349	46.265	120	120
3	3.571	5.357	5.357	25	35.714	75	75
4	2.813	14.063	2.813	11.25	14.063	45	45
5	0	28.235	21.176	35.294	35.294	120	120
Current Destinations Total	43.613	95.547	135.611	103.894	131.336
Destinations Total Future year	48	75	48	150	189

Doubly Constraint Growth Factor Matrix For Future Year

Zone	1	2	3	4	5	Current Origins Total	Origins Total Future year
1	37.598	46.202	27.301	35.716	0	146.817	150
2	5.671	1.91	13.166	25.403	75.058	121.207	120
3	2.794	2.823	0.927	26.962	42.412	75.919	75
4	1.937	7.83	0.964	15.022	19.49	45.244	45
5	0	16.234	5.643	46.897	52.04	120.814	120
Current Destinations Total	48	75	48	150	189
Destinations Total Future year	48	75	48	150	189

References Books

• J. De D. Ortuzar and L.G. Willumsen (1996), Modelling Transport. Wiley Publications, London.
• C. S. Papacostas and P. D. Prevedouros (2001), Transportation Engineering & planning. Prentice-Hall of India, New Delhi.
• L. R. Kadiyali. Traffic Engineering and Transport planning. Khanna Publications, Delhi.